2.6 A Sample Problem
The length of time X, needed by students in a particular course to complete a 1 hour exam is a random variable with PDF given by
For the random variable X,
- Find the value k that makes f(x) a probability density function (PDF)
- Find the cumulative distribution function (CDF)
- Graph the PDF and the CDF
- Find the probability that that a randomly selected student will finish the exam in less than half an hour
- Find the mean time needed to complete a 1 hour exam
- Find the variance and standard deviation of X
The given PDF must integrate to 1. Thus, we calculate
Therefore, k = 6/5. Notice also that the PDF is nonnegative everywhere.
The CDF, F(x), is the area function of the PDF, obtained by integrating the PDF from negative infinity to an arbitrary value x.
If x is in the interval (-∞, 0), then
If x is in the interval [0, 1], then
If x is in the interval (1, ∞) then
The CDF is therefore given by
The PDF and CDF of X are shown below.
The probability that a student will complete the exam in less than half an hour is Pr(X < 0.5). Note that since Pr(X = 0.5) = 0 (since X is a continuous random variable) it is equivalent to calculate Pr(x ≤ 0.5). This is precisely F(0.5):
The mean time to complete a 1 hour exam is the expected value of the random variable X. Consequently, we calculate
To find the variance of X, we use our alternate formula to calculate
Finally, we see that the standard deviation of X is